Suggestions to make a "uwrap" animation with this ball model
If you are looking for result like in your second image - ready to use file (print-cut-glue), use build in "Paper Model Export" addon by Emu.

But for unfolding, use his "Convert to Armature" addon - download version for 2.8x. More info here.



Unfold Animation


download-install-enable addon
select object, go to edit mode (Tab), select edges you want split
Ctr+E > Mark Seam
search Split Seams
additionally select one face you prefer to start generate bone tree
search Convert to Armature


That's all ... to be precise, object is not animated yet. Script generates Armature object with bones assigned to faces and creates unfolded Pose. To animate unfolding - go to Pose Mode (Ctrl+Tab), select all bones, add Keyframe (I) > Rotation, move on Timeline, Clear Rotation (Alt+R), add second Keyframe > Rotation.



If you have more complex mesh you can let "Export Paper Model" script to Mark Seams for you (3Dview > Properties Panel (N) > Paper > Unfold). Continue as described above - search Split Seams ...